MHT-CET : Mathematics Entrance Exam

MHT - CET : Mathematics - Parabola Formulae Page 1

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Parabola

Def: Parabola is the locus of a point on a plane, which is equidistant from a fixed point and a fixed line of that plane.
If
P is the variable point, S is the fixed point and PM is the distance from a fixed line 'd', then PS = PM (Focus-Directrix property)

Standard form:
y2 = 4ax, y2 = - 4ax, x2 = 4ay, x2 = - 4ay
Important information:

 y2 = 4ax y2 = - 4ax x2 = 4ay x2 = - 4ay Co-ordinate  of Vertex (0, 0) (0, 0) (0, 0) (0, 0) Co-ordinate  of focus (a, 0) (- a, 0) (0, a) (0, - a) Axis x-axis x-axis y-axis y-axis Equation of directrix x + a = 0 x - a = 0 y + a = 0 y - a = 0 Length of Latus rectum 4a 4a 4a 4a Co-ordinate of the end  of Latus rectum (a, 2a) (a, - 2a) (- a, 2a) (- a, - 2a) (2a, a) (- 2a, a) (2a, - a) (- 2a, - a) Parametric Co-ordinate x = at2, y = 2at x = - at2, y = 2at x = 2at , Y = at2 x = 2at, y = - at2 Equation of axis y = 0 y = 0 x = 0 x = 0 Extent(Quadrants) I & IV II & III I & II III & IV

General Equation: (y - b)2 = 4a(x - a)

Important information:

 Co-ordinate  of Vertex Co-ordinate  of focus Equation of directrix Length of Latus rectum Co-ordinate of the end  of Latus rectum Equation of axis Parametric Co (a, b) (a + a,b) X = a + a 4a (a + a, 2a + b) (a + a, - 2a + b) y = b x = at2 + a, y = 2at + b

Tangent to a parabola:

Equation of tangent to a parabola y2 = 4ax at (x1, y1) is yy1 = 2a(x + x1)
Equation of tangent to a parabola
y2 = 4ax at (at2,2at) is yt = x + at2.

General equation of tangent to a parabola y2 = 4ax in terms of slope is y = mx

 a m

General equation of tangent to y2 = 4ax in terms of parameter 't' is yt = x + at2.

Condition of tangency c =

 a m

Point of contact is

(

-

 a m2

,

 a m

)

Ellipse:

Def: Ellipse is the locus of a point on a plane such that the ratio e of its distances from a fixed point and a fixed line of that plane is constant where e <1.
If
P is the variable point, S is the fixed point and PM is the distance from a fixed line 'd' then PS = e PM (Focus-Directrix property)

Standard form:

 x2 a2

+

 y2 b2

= 1(a > b)

 x2 a2

+

 y2 b2

= 1(a > b)

Co-ordinate of Vertex

(0, 0)

Co-ordinate of focus

(± ae, 0)

Major axis

x-axis

Length of Major axis

2a

Minor axis

y-axis

Length of Minor axis

2b

Equation of directrix

x ±

 a e

= 0

Length of Latus rectum

 2b2 a

Co-ordinate of the end of Latus rectum

(

± ae, ±

 b2 a

)

Relation between a, b and e

b2 = a2(1 - e2)

Parametric Co-ordinate

x = a cosq, y = b sinq

Tangent to an Ellipse:

Equation of tangent to an ellipse

 x2 a2

+

 y2 b2

= 1 (a > b) at (x1, y1) is

 xx1 a2

+

 yy1 b2

= 1

Equation of tangent to an ellipse

 x2 a2

+

 y2 b2

= 1 at (a cosq, b sinq) is

 xcosq a

+

 ysinq b

= 1

General equation of tangent to an ellipse

 x2 a2

+

 y2 b2

= 1 in terms of slope m is

y = mx

 Condition of tangency c =

Point of contact is

(

±

 m

,

)

Hyperbola:

Def: Hyperbola is the locus of a point on a plane such that the ratio e of its distances from a fixed point and a fixed line of that plane is constant where e >1.
If
P is the variable point, S is the fixed point and PM is the distance from a fixed line 'd' then PS = e PM (Focus-Directrix property)

Standard form:

 x2 a2

-

 y2 b2

= 1(a > b)

Important information:

 x2 a2

-

 y2 b2

= 1(a > b)

Co-ordinate of Centre

(0,0)

Co-ordinate of focus

(± ae, 0)

Major axis

x-axis

Length of Major axis

2a

Minor axis

y-axis

Length of Minor axis

2b

Equation of directrix

X ±

 a e

= 0

Length of Latus rectum

 2b2 a

Relation between a,b and e

b2 = a2(e2 - 1 )

Co-ordinate of the end of Latus rectum

(

± ae, ±

 b2 a

)

Parametric Co-ordinate

X = a secq, y = b tanq

Tangent to a Hyperbola

Equation of tangent to a hyperbola

 x2 a2

-

 y2 b2

= 1 (a > b) at (x1, y1) is

 xx1 a2

-

 yy1 b2

= 1

Equation of tangent to a hyperbola

 x2 a2

-

 y2 b2

= 1 at (a secq, b tanq) is

 x secq a

+

 y tanq b

= 1

General equation of tangent to a hyperbola

 x2 a2

-

 y2 b2

= 1 in terms of slope m is

y = mx

 Condition of tangency c =

Point of contact is

(

 m

)

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